2D vs 3D clustering of the elliptic particulates: The correlation with the percolation thresholds
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1.
Boğaziçi University
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2.
California Institute of Technology
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3.
American University of Beirut
Abstract
We develop a continuum percolation procedure for the aggregation of the elliptic fillers in the 2 and 3-dimensional media. Given random distributions for the locus and rotations of the elements with a specified original density p, each medium achieves chains of elements through overlapping to achieve a connection density of ρ. In this regard, typically 3D aggregation is more efficient than 2D due to the possibility of additional connectivity from the in/out (i.e. depth) directions. Hence, when increasing the number of fillers the 3D percolation system experiences an early increase in the connection density ρ, which typically occurs in the neighborhood of the percolation threshold p_c. We initially develop a new iterative method to compute the percolation threshold p_c in finite systems. Subsequently, we show that such early divergence between 2D-3D percolation systems is followed by a later convergence stage, as the number of fillers progressively increases. Consequently, we show, conceptually and computationally, that the maximum 2D-3D difference in the connections density Δ_ρ_max correlates directly with the respective 2D-3D difference in the percolation thresholds Δ_p_c , where a large pool of computational samples were generated by varying the aspect ratio as well as the relative scale of the particles. The results and respective analyses could be useful for the design of binary composite membranes of a specified thickness (i.e. thin → 2D, thick → 3D) for achieving the desired homogenized physical property.
Copyright and License
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Contributions
Asghar Aryanfar: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Mahmoud Yamani: Writing – original draft, Validation, Software, Formal analysis, Data curation. William A. Goddard: Supervision, Project administration.
Conflict of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
No funding or acknowledgements information available
Supplemental Material
MMC 1. Computes the maximum 2D-3D difference in the connected density, when the percolation thresholds are specified.
MMC 2. Computes the overlap possibility of any two particles and provides the adjacency matrix ; 0: separated, 1: connected.
MMC 3.
MMC 4. Computes the density of the 3D connected cluster by counting the grids points falling within.
MMC 5. Generates the elliptical particles with random allocation and rotations in 3D.
MMC 6. Visualizes the 3D medium of the connected (color) and disconnected (gray) particles.
MMC 7. Establishes the 3D connected cluster by propagating from the prescribed central particle, through the overlapping elements. As the result the particles belonging to the connected cluster are extracted out.
MMC 8. Finds the percolation threshold in 3D domain with the assigned geometry of the particles, through convergence of the upper and lower bounds.
MMC 9. Generates randomized ellipses in 2D for a range of specified particle geometry, and finds the density of the connected particles.
MMC 10. Computes the density of the 2D connected cluster by counting the grids points falling within.
MMC 11. Computes the density of the 2D connected cluster by counting the grids points falling within (optimized relative to Dens2 in vectorial form to run faster).
MMC 12. Generates the elliptical particles with random allocation and rotations in 2D.
MMC 13. Computes the correlation between the normalized connected density versus the relative variation in the number of particles and interpolates the into a quadratic polynomial.
MMC 14. Visualizes the 2D medium of the connected (color) and disconnected (gray) particles.
MMC 15. Establishes the 2D connected cluster by propagating from the prescribed central particle, through the overlapping elements. As the result the particles belonging to the connected cluster are extracted out.
MMC 16. Finds the percolation threshold in 2D domain with the assigned geometry of the particles, through convergence of the upper and lower bounds.
Additional details
Dates
- Available
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2025-02-27Version of record